My
first search was to look at musical practice in ancient times, not
only in Europe but all over the world. There were several other
musical scales besides the diatonic scale, where the semitones were
located in other places than from me-fa and si-do. In the Gregorian
modals, for instance, the different placement of the semitones
creates the specific modes.

Pentatonic
Music

In pentatonic folk music
semitones do not exist. By practical experience, people have found
out that the five-note scale allowed the possibility of playing in
any key without significant disharmony. Theorists would say that the
scale was composed of ascending and descending fifths, only in two
steps in each direction. A pentatonic scale can be played by only
using the black keys on the piano.

EASTERN
MUSIC

After a study of ancient
main cultural music, mainly Chinese and Indian, I realized how
universal the concept of the octave was in every musical
culture.According to Helmholtz, the Arabic and Persian scales,
and the Japanese and the Pacific scales are also within an octave.
However, the division of the octave differs from culture to
culture.Arab music divides the octave into sixteen unequal
intervals. The Persians divided their octave into 24 steps, so they
must have used quarter tones. From excavated Egyptian flutes, a seven
note scale C, D, E, F#, G, A, B, has been discovered, which is
identical with the Syntolydian scale of ancient Greece. Japanese
music used mainly a pentatonic scale.

Chinese
music

Music was the
cornerstone of the Chinese civilization, the longest living culture
in history. It was considered to embody within its tones elements of
the celestial order. The audible sound, including music, was but one
form of manifestation of a much more fundamental form of
Super-physical Sound. The fundamental Primal Sound was synonymous
with that which the Hindus call OM. The Chinese believed that this
Primal Sound, Kung or ‘Huang Chung’ (directly translated
‘yellow bell’) was, though inaudible, present everywhere
as a Divine Vibration.Furthermore, it was also divided into 12
lesser Sounds or Tones. These twelve Cosmic Tones were emanations of,
and an aspect of, the Primal Sound, but were closer in vibration to
the tangible, physical world. Each of the 12 Tones was associated
with one of the 12 zodiacal
regions of the heavens.

Audible sound was
conceived as being a physical level manifestation of the 12 tones.
Sound on Earth was a kind of sub-tone of the celestial vibration. It
was believed to contain a little part of the celestial tones' divine
power.“As above, so below”, as the Egyptian Hermes
Thot said. In the Lord’s Prayer, a similar wish is spoken.

For the ancient Chinese,
the alignment with the divine prime tone
was the Emperor's most important task. The alignment of earth with
heaven, and man with the Supreme, was literally the purpose of life.
The entire order and affairs of the State were dependent upon the
right tuning of the fundamental tone, the ‘yellow bell’,
or Kung.

As an ancient text
warns: “If the Kung is disturbed, then there is
disorganization; the prince is arrogant.” If the Kung was
out of tune, because the celestial realm has changed, disorder and
inharmonious behavior in society became obvious. Every instrument
(including measuring instruments) was tuned and utilized in
accordance with the holy tone.

The instrument that
could give to man the fundamental tone for a musical scale in perfect
harmony with the universe was the key to earthly paradise, and
essential to the security and evolution of society.It became the
Chinese Holy Grail.One
legend tells of the amazing journey of Ling Lun, a minister of the
second legendary Chinese Emperor, Huang Ti. Ling Lun was sent like an
ancient Knight of King Arthur to search for a special and unique set
of bamboo pipes. These pipes were so perfect that they could render
the precise standard pitches to which all other instruments
throughout the land could be tuned.

That sacred tone, which
relates to the Western modern pitch of F, was considered as the
fundamental cosmic tone. The Chinese were aware of the slow changing
cosmic influence, and consequently the Kung has to change
accordingly. The Emperor had the task of tuning the Kung so it was in
alignment with the cosmic tone.

Tuning
the Sacret Kung

Cousto has in his book
The Cosmic Octave an interesting observation on this matter. He
relates the Kung to the frequency of the
Platonic Year. The duration of the Platonic Year, (The
Pythagorean Great Year) is about 25,920 years and represents the
amount of time the axis of the Earth takes to complete a full
rotation.

The vernal equinox is
the point at which the equator (of Earth) intersects the ecliptic (or
zodiac), which is the position of the sun at the beginning of spring
- March 21st.The vernal equinox takes an average of 2,160 years
to travel through one sign of the zodiac. This period of time is
known as an age. It is not possible to state exactly when one age is
ending and a new beginning, because the signs overlap to a certain
degree. The journey of the vernal equinox through each of
the 12 signs of the Zodiac equals one great year of approximately
25,920 years. (Presently we are on the cusp of Aquarius as the age of
Pisces is ending). This number of years is close to the high
number of generating
fifths when we come into a cycle of 25,524 notes.

Cousto
calculates the note of the Platonic year to be F in the Western Equal
Temperament pitch, which is found in the 48th octave with a frequency
of 344.12 Hz., or in the 47th octave to be 172.06 Hz. Note that
the corresponding a' has a frequency of 433.564 Hz. (Modern Western
concert pitch is 440 Hz.)Calculation:
31 556 925.97(the tropical year in seconds) * 25,920 (Platonic year).
Since the length (of a vibrating string, or the period of time) is in
reverse proportionality to the frequency, the length of the Platonic
year in seconds shall be the denominator. The frequency is very low,
so we will raise the frequency to the range of hearing by multiplying
with the necessary amount of octaves, e.g. 48 octaves so we arrive to
344,12 Hz. (47 octaves will be the half, 172,06 Hz.)If we want to
reach the spectrum of light, we multiply with 89 octaves which leads
us to a frequency of 1/31 556 925.97 * 1/25 920 * 2 89 = 7,56 * 10 14
Hz. corresponding to a wavelength of 0.396 micrometer, which we
perceive as violet near the ultra violet. This is the color of the
Platonic Year. The complementary color to violet is yellow. Their
fundamental tone was called the yellow bell.

It is a wonder for me
how the ancient Chinese could be aware of their sacred fundamental
tone, Kung, being in accordance with the Platonic Year, and choose
the great rhythm of the Earth.

Creation
of a scale

It might be a surprise
that the diatonic scale was the foundation for the ancient Chinese
and the Indian music, though the musical theory and practices differ
from the Western.

For the old Chinese,
their musical scale was developed by the circle of perfect fifths up
to 60 degrees or keys, the 60 Lü, though they usually only used
the first 5 fifths in their pentatonic music, because they knew that
these represent the limit of consonance in modal music. In addition,
the ancient Chinese saw a symbolic representation in the pentatonic
scale, rooted in their belief in music as being the representation of
the relationship between heaven and earth (the five elements).

The Chinese were well
aware centuries ago of the existence of our modern Equal Temperament.
They dismissed such a tempered scale not only for its badly false
notes, but mainly because the tuning was not in alignment with the
cosmic tone.According
to the book by David Taime, The Secret Power of Music, 3 was the
symbolic numeral of heaven and 2 that of the earth; sounds in the
ratio of 3:2 will harmonize heaven and earth. As a way to apply that
important concept, the Chinese took the foundation note, ‘Huang
Chung’, and from it produced a second note in the ratio of 3:2.

A more in-depth
explanation made by Alain Daniélou in his ”Music and the
Power of Sound”:“Music, being the representation of
the relationship between heaven and earth, must quite naturally have
this confirmation of a center or tonic (gong) surrounded by four
notes assimilated to the four directions of space, the four
perceptible elements, the four seasons, and so on. "

“The pentatonic
scale thus presents a structure that allows it to be an adequate
representation of the static influence of heaven on earth. But a
static representation of a world in motion could not be an instrument
of action upon that world. It is necessary to evolve from the
motionless to the moving, from the angular to the circular, from the
square to the circle. To express the movements of the universe, the
sounds will have to submit to the cyclic laws that, in their own
field, are represented by the cycle of fifths.”

The
spiral of fifths

As
we have already seen, the fifth is the third sound of the series of
harmonics, the first being the fundamental and the second its octave.
According to the formula of the Tao-te ching, “One has produced
two, two has produced three, three has produced all the numbers”,
we can understand why the third sound, the fifth, must necessarily
produce all the other sounds by its cyclic repetitions. ←Observe
the feminine & masculine notes respectively pink & blue.

The first to be produced
will be the four principal sounds, which form comparatively simple
ratios with the tonic. For
the sake of convenience we will use Western notes: See Chinese
& Western Music.I,
C II, G = 3/2 III, D = 9/8 = (3/2)2* ½ (lower an
octave) IV, A+ ( a comma sharp) = 27/16 = (3/2)3* ½ (lower an
octave)V, E+ (a comma sharp) = 81/64 = (3/2)4* ¼ (lower 2
octaves).These five primart
sounds represent the elementary structure of the perceptible world,
the pentatonic scale. These sounds are used in music, as you can play
the five black keys on the piano. Howevwer, the next two fifths have
to be added as two auxiliry sounds: VI,
B+, (a comma sharp) = 243/128 = (3/2)5* ¼ (lower 2
octaves)VII, L+F? (sharpen a major half tone) = 729/512 =
(3/2)6* 1/8 (lower 3 octaves).

The
seven-notes Chinese scale

C

D

E+

(F)¤L+F#1/1

G

A+

B+

C'

1/1

9/8

81/64

4/3
(729/512)

3/2

27/16

243/128

02/01/13

¤Let us note here that the most striking
difference between the system of fifths and that of harmonic
relations to a tonic, resides in the perfect fourth, which is an
essential interval in the scale of proportions, but in the scale of
fifths it is an augmented fourth as its sixth fifth, (3/2)6..

The two auxiliary sounds
– 243/128 and 739/512 – should not be used as
fundamentals, though they are needed for
transpositions, because they belong to
the scale of invisible worlds, and
therefore we can neither perceive their accuracy nor build systems
upon them without going out of tune.Instead of starting from C,
we could have begun one fifth below, that is to say, from F, and we
would have obtained this essential note without changing anything in
our scale, except that, since we begin with a masculine
interval instead of a feminine interval,
the character of the whole system is modified.

The five successive
fifths, whether in an ascending or a descending series, represent the
limit of consonance in modal music too. Beyond this limit, no
interval can appear harmonious, nor can it be accurately recognized.
A rule originating from the same principle was also known in medieval
Europe, where the tritone was prohibited as diabolical, that is, as
connected with forces that are supernatural and therefore
uncontrollable. Folk music in its pentatonic form had
understood this too by only using the span of two fifths up and
down.After
these seven notes, the next five notes generated by the series of
fifths are: VIII, bDb lowered a minor half tone, IX,
bAb lowered a minor half tone, X, bEb a minor half tone lower, XI,
bBb a minor half tone lower, XII, F+ a comma sharp.

We now have twelve
sounds, which divide the octave chromatically into twelve half
tones. The twelfth fifth (note 13) in
a 7 octave span brings us back to the fundamental, but with a slight
difference.It is higher than the fundamental by one comma,
the Pythagorean comma (312 / 219 = 531,441/524,288, (5.88 savarts or
23.5 cents). It is, therefore, in our notation, C+, one comma
sharp.In this way, successive series of twelve fifths will be
placed one above the other at one-comma intervals, up to the 52nd
fifth (note 53) which fill the octave.

The Chinese continued
the cycle of fifths up to 25,524 notes, with a basic interval of
0.0021174 savarts. This cycle is very near to that of the
precession of the equinoxes,or
the Pythagorean Great Year, which is of 25,920 solar years. Why
the Chinese continued so many octaves in the cycle of fifths could
have something to do with their reference tone, Kung.

In practice, for reasons
that are symbolic as well as musical, after the 52nd fifth (53rd
note) the Chinese follow the series only for the next seven degrees,
which place themselves above those of the initial seven-note scale,
and they stop the series at the 60th note. The reason given is that
12 (the number of each cycle) * 5 (the number of the elements) = 60.

The
scale of 60 Lü

The Chinese scale, being
invariable, constitutes in effect a single mode. Every change in
expression will therefore depend upon modulation, a change of
tonic.Firstly, the choice of gender: fifths whose numbers in the
series are even are feminine. The odd numbered fifths are
masculine.The choice of tonic is dependent on complicated rules
and rituals, whose main purpose is to be in accordance with celestial
as well as earthly influx or circumstances. Accordingly, the
Chinese have to choose the right key for the hour of the day and the
month, even during a performance. It is an extensive scheme, but
to get an idea we can say that it corresponds to political matters,
seasons, hour of the day, elements, color, geographic direction,
planets and moon. This scale of fifths, perfect for transposition
because of its extreme accuracy, also allows the study of
astrological correspondences and of terrestrial influx in their Tone
Zodiac.

We notice that the
Chinese scale is very similar to the Pythagorean tuning, which was
also produced by generating a perfect fifth (3:
2). When the Chinese derived their scale goes back to 3000 BC,
when European stone-age man was still beating wooden logs. The
prevalent opinion in the West about our music superiority should
hereby be moderated.

The
Indian music system

The ancient Indians had
a less formalized approach to their music than the Chinese. Generally
speaking they emphasized the personal inner contemplation more than
the outward organized rituals. One can say that they sought inner
alignment with the divine supreme by means of the sounds AUM or OM,
which were (are) the earthly sound of the prime creator, Brahman.

For the Hindus, as the
Chinese, the spoken or chanted words were the carrier of some of the
creative energy, and composed by the prime Creator. Pronounced
correctly, it was believed that special words were able to alter
humans’ thoughts and feelings and literally change and form
physical matter.

Raga is the basic form
in classical Indian music. There is a whole system of Ragas, which
differ respectively between North and South India. Originally there
were only 7 Ragas. These may have been the remnant of an ancient
reference to the seven Cosmic Tones: the seven principal notes, or
savaras, connected with the seven main planets, and two secondary
notes corresponding to the nodes of the moon. This brings the total
number of notes in the scale to nine principal notes, which is
related to the nine groups of consonants of the Sanskrit alphabet.

The Raga system grants
musicians freedom of expression within the
limitations of a certain inviolable mode. Since music was so
important a force in altering phenomena upon Earth, they considered
it would be unwise, dangerous, and perhaps even suicidal in the long
run to allow musicians to perform whatever they wished.

The Indian solution was
then to apply a system of rules which, while effectively determining
what type of music was performed and even its spiritual
atmosphere and the period of the day, did not indicate the notes
themselves. This was a convincingly successful solution to the
problem that the music of ancient civilizations always came up
against.

The
Chinese had a more rigid system. They created variations by use of
instruments, and especially in the expression of the single note. The
dimensions of tone color, or timbre, were highly developed in the
East. The ear had to learn to distinguish subtle nuances. The same
note, produced on a different string, has a different timbre. The
same string, when pulled by different fingers, has a different
timbre, etc. Furthermore, and very important, the
whole spiritual being of the musician himself was crucial.
That applies also to Indian music.

As in the Western
diatonic scale, the Indian scale was based on 7 main notes: SA, RE,
GA, MA, PA, DHA and NI. If we go back to the most ancient texts on
music, the scales were divided into two
tetrachords, similar to the ancient Greeks, and later put
together with a whole tone (9/8) between, Ma – Pa, so a full
octave was completed. The Indian notes relate broadly to the
Western ratios, though the tuning is very harmonious and creates a
world of difference. We have to emphasize that the use of harmony as
we know it was, but is no longer, musically practised.

Here is a crucial point.
The Indian music is modal. There is
a strong relationship to the tonic. When a third is played it always
relates to the third degree; whereas in Western harmonious tradition
the third has a relative position, because it can be the root, the
fifth or third of a chord. Eastern listeners often make
remarks such as: “Beethoven symphonies are interesting, but why
have all those chords been introduced, spoiling the charm of the
melodies?”The modal music of India is 'horizontal'
as the Western is 'vertical'. The
vertical, harmonious system, in which the group of related sounds is
given simultaneously, might be more direct though also less clear.
The accurate discrimination of the different elements that constitute
a chord is not usually possible.

The modal, horizontal
system, on the other hand, allows the exact perception and immediate
classification of every note, and therefore permits a much more
accurate, powerful and detailed outlining of what the music
expresses.One can say that the
attention span in
the Eastern musical language has to be much longer since, in time,
the different and distinct sounds adding up in the listeners mind
create the chords or the whole musical idea. Only then, by
remembering with attention all the elements that constitute the
musical image, can the full meaning finally be understood. The
Indian musical system operates with a combination of fixed and
mutable pitch, so the key can be recognized along with variable
notes. The 2nd, 3rd, 4th, 6th and 7th notes are variable, but the 1st
(Sa or Do) and the perfect 5th (Pa or Sol) are immutable and of a
fixed pitch. The drone is accordingly often Do-Sol (Sa-Pa), which
becomes the ultimate open chord containing all other notes within it
as a series of subtle harmonics.This drone (a constant note or
tonic), whether actually played on an instrument like the tampura or
simply heard within oneself as the Om sound, is the constant
reference without which no Indian musician would play.

One must not be
confused by the vast use of micro intervals, sliding or bending
the notes, prominent in Indian music. The musicians can freely use
these microtones as private points, often moving freely between two
notes as a kind of infinitely exploitable space, eventually returning
home to the tonic of the Raga. The musician has a freedom to play
tones as his inspiration demands so long as he obeys the sacred rules
of types and its mood.

The
22 Shrutis (degrees)

Musical intervals can be
defined in two ways, either by numbers (string lengths, frequencies)
or by their psychological
correspondences, such as feelings and images they necessarily evoke
in our minds. There is no sound without a meaning, so the Indians
consider the emotions that different
intervals evoke as exact as sound ratios. The feeling of the shrutis
depends exclusively on their position in relation to the tonic, and
indicates the key for the ragas.

The 22 different keys or
degrees encompass what the Indians consider the most common feelings
and reflections of the human mind. They were aware of the division of
the octave into 53 equal parts, the Pythagorean Comma, and its
harmonic equivalent, the comma diesis, (the syntonic comma, the
difference between the major and the minor tones). However, they
chose the 22nd division of the octave for reasons based on the limit
of human ability to differentiate the keys, as well as for
psychological and metaphysical reasons. The symbolic
correspondences of the numbers 22 and 7, (7 strings and main notes),
could also play a part since the relationship between the circle and
the diameter is expressed as the approximate value of Pi, 22/7.

The
modal or Harmonic division of the octave

Indian music is
essentially modal, which means that the intervals on which the
musical structure is built are calculated in relation to a permanent
tonic. That does not mean that the relations between notes other than
the tonic are not considered, but that each note will be established
first according to its relation to the fixed tonic and not, as in the
case of cycle of fifths, by any permutations of the basic note.

The modal structure can
therefore be compared to the proportional
division of the string (straight line) rather than to the periodic
movement of the spiral of fifths.All the notes obtained in the
harmonic system are distinct from those of the cyclic system, which
is based on different data. Though the notes are theoretically
distinct and their sequence follows completely different rules, in
practice they lead to a similar division of the octave into
fifty-three intervals.

The
scale of proportions is made of a succession of syntonic
commas, 81/80, which divide the octave into 53 intervals.
Among those, 22 notes were chosen for their specific emotional
expressions:

Note degree

Interval

Value in cents

Interval Name

Expressive qualities

1

1/1

0

unison

marvelous,
heroic, furious

2

256/243

90.22504

Pythagorean
limma

comic

3

16/15

111.7313

minor
diatonic semitone

love

4

10/9

182.4038

minor
whole tone

comic,
love

5

9/8

203.9100

major
whole tone

compassion

6

32/27

294.1351

Pythagorean
minor third

comic,
love

7

6/5

315.6414

minor
third

love

8

5/4

386.3139

major
third

marvelous,
heroic, furious

9

81/64

407.8201

Pythagorean
major third

comic

10

4/3

498.0452

perfect
fourth

marvelous,
heroic, furious

11

27/20

519.5515

acute
fourth

comic

12

45/32

590.2239

tritone

love

13

729/512

611.7302

Pythagorean
tritone

comic,
love

14

3/2

701.9553

perfect
fifth

love

15

128/81

792.1803

Pythagorean
minor sixth

comic,
love

16

8/5

813.6866

minor
sixth

comic

17

5/3

884.3591

major
sixth

compassion

18

27/16

905.8654

Pythagorean
major sixth

compassion

19

16/9

996.0905

Pythagorean
minor seventh

comic

20

9/5

1017.596

just
minor seventh

comic,
love

21

15/8

1088.269

classic
major seventh

marvelous,
heroic, furious

22

243/128

1109.775

Pythagorean
major seventh

comic,
love

The
Ancient Egyptians

The ancient Egyptians
had similar beliefs to the Chinese and Hindus. In their Book of the
Dead and other sources, it is stated that God, or his lesser servant
gods, created everything, by combining visualization with utterance.
First the god would visualize the thing that was to be formed; then
he would pronounce its name: and it would be.

From as late as the
reign of Alexander II, a text dating from about 310 BC still has the
God of Creation, Ra, declaring: “Numerous are the forms from
that which proceeded from my mouth.” The god Ra was also called
Amen-Ra, with the prefix ‘Amen’. The Egyptian priesthood
understood well the word Amen, or AMN, and it was equated with the
Hindu OM.

Egyptian music, as does
Greek, most probably had its roots in Indian music, or at least in
that universal system of modal music whose tradition has been fully
kept only by the Indians. The pyramid can easily be a
symbolic representation of Earth with its four perceptible elements,
and all its characteristics that are regulated by the number four –
the four seasons, four directions of space, etc.; especially the
projection of the single into the multiple.

WESTERN
MUSICPythagoras

The Greek philosopher
Pythagoras (570 - 490 BC) spent 22 years in Egypt, mainly with the
high priest in Memphis, where he became initiated into their secret
knowledge of Gods. When the Persians conquered Egypt, he was kept in
captivity in Babylon for sixteen years before he could return to
Greece and begin his teaching. I began to study the theory
of the Pythagoreans and their esoteric schools. Very little is known
of them. Pythagoras demanded silence about the esoteric work. This
historic school was founded in the Greek colony Kroton, in southern
Italy, about 2,500 years ago.

I realized after reading
dozens of books about the matter what an outstanding role that school
played in the establishment of western civilization. He created an
entirely new concept. Any person - man or woman
- who had a sincere wish for knowledge could enter the school
stepwise, with a number of initiations. The tradition of a
priesthood’s monopoly of knowledge of God was broken.

Pythagoras' study of the
moving string and his discovery of the harmonic progression of simple
whole numbers was the first real scientific work and creation of
modern science. But his vision went far beyond present science in his
deep understanding of the integration of the triad: A
– science, B – work on being, C – love and study of
God. Something modern science could
learn from!

Nicomachus
of Gerasa

Nicomachus the
Pythagorean (second century B.C.) was the first who wrote about
Pythagoras’ legendary encounter with ‘the harmonious
blacksmith’ and the weights of the 4 different hammers
being 12, 9, 8 and 6, that determined the variation in the pitches
Pythagoras heard. This story illustrates how the numerical
proportions of the notes were discovered. His methodical measuring of
the hammers and how the sound was produced and related (collecting
data), then making experiments with strings, their tension and
lengths (repeating the findings and, with mathematics, formulating
them into a law), was the first example of the scientific method.

We will not dwell on the
question of the force of the impact or the tension of the strings,
which later was discovered as the square root of the force, but just
stick to the proportion of weights and the pitches he heard, which
led him to his discovery.Pythagoras' experiments led to the
combination of two tetrachords, (two fourths),
separated with a whole tone, 9/8, which constitute an octave.
He changed the traditional unit in Greek music, the tetrachord, into
the octave by an octachord.

In the time of
Pythagoras the tradition was strongly based on the seven strings of
the lyre, the heptachord. The Greeks considered the number 7 sacred
and given by the god Hermes, who handed down the art of lyre playing
to Orpheus. The seven-string lyre was also related to the seven
planets, amongst other things the ancients venerated. The lyre
often, but not always, consisted of seven strings comprising two
tetrachords, each one spanning the most elementary concord, the
fourth, both joined together on the note mese.

According
to legend, a son of Apollo, Linos, invented the four-stringed lyre
with three intervals, a semitone, whole tone and a whole tone
comprising a fourth; the fourth, ‘the first and most elementary
consonance’ as Nicomathus calls it, and from which all the
musical scales of ancient Greek music eventually developed.Trepander
of Antissa on Lesbos, born about 710 B.C., assumed a mythological
status for his musical genius. His most lasting contribution was
perhaps his transformation of the four-stringed lyre to the
instrument which became institutionalized by tradition to the
heptachord.Trepander did before Pythagoras extend the heptachord
from its minor seventh limits to a full octave, but without having to
add the forbidden eighth string.He removed the Bb string, the
trite of the conjunct tetrachord, and added the octave string, E1,
yielding a scale of E F G A C D E1.This arrangement left a gap of
a minor third between A and C, and seemed to have enhanced the Dorian
character of Trepander's composition.

Harmonia

Only Pythagoras escaped
censure for adding an eighth string to the ancient and venerated lyre
because of his position as a great master and religious prophet. His
purpose was to teach man the unifying principle and immutable laws of
harmonia by appealing to his highest powers - the rational intellect
and not to his untrustworthy and corruptible senses. Pythagoras
altered the heptachord solely to engage man's intellect in proper
‘fitting together’ - harmonia
- of the mathematical proportions.

Plutarch (44-120 B.C.)
states that for Pythagoras and his disciples, the word harmonia meant
‘octave’ in the sense of an attunement which manifests
within its limits both the proper fitting together of the concordant
intervals, fourth and fifth, and the difference between them, the
whole tone. Moreover, Pythagoras proved that whatever can be
said of one octave can be said of all octaves. For every octave, no
matter what pitch range it encompasses, repeats itself without
variation throughout the entire pitch range in music. For that
reason, Pythagoras considered it sufficient to limit the study of
music to the octave.

This means that within
the framework of any octave, no matter what its particular pitch
range, there is a mathematically ordained place for the fourth, the
fifth, and for the whole tone. It is a mathematical matter to show
that all of the ratios involved in the structure of the octave are
comprehended by the single construct: 12-9-8-6. For
the Pythagoreans, this construct came to constitute the essential
paradigm - of unity from multiplicity.

The
arithmetic and harmonic mean

We see that 12:6
expresses the octave, 2:1; 9 is the arithmetic
mean, which is equal to the half of the sum of the extremes,
(12 + 6)/2 = 9. Further, 8 is the harmonic
mean of 12:6, being superior and inferior to the extremes by
the same fraction. Expressing this operation algebraically,
the harmonic mean is 2ac/a+c, or in this series, 2*12*6/12+6 = 8.

Among the peculiar
properties of the harmonic proportion is the fact that the ratio of
the greatest term to the middle is greater than the middle to the
smallest term: 12:8 >8:6. It is this property that made the
harmonic proportion appear contrary to the arithmetic proportion.

In terms of musical
theory, these two proportions are basic for division of the octave
since the fifth, 3/2, is the arithmetic
mean of an octave and the fourth, 4/3, is the harmonic
mean of an octave.The principle of dividing the string by an
arithmetical proportion is done by the formula: a:b is divided by
2a:(a+b) and (a+b):2b.

The ancient Greeks
presumably did such division in their studies of the singing string
of the monochord.

The
semitone

We have already seen
that in the diatonic genus each tetrachord was divided into two full
tones and one semitone. A full tone derives from a fifth minus a
fourth, 3/2 - 4/3 = 9/8. The semitone will be 4/3 - (9/8 + 9/8), or
4/3 - 81/64 = 256/243. This semitone is called leimma, and
is somewhat smaller than the half tone computed by dividing (for
musical ratios dividing means the square root) the whole tone in
half: (9/8)½ = 3/2*2½.

The square root of 2 was
for the Pythagoreans a shocking fact, because their concept of
rational numbers was shattered. (For me it represents the beauty of
real science, because it revealed the flaws in the Pythagorean
paradigm of numbers). Their own mathematic proved with the
Pythagoreans’ doctrine of the right-angle triangle (the sum of
the squares of the two smaller sides of a right-angled triangle is
equal to the square of the hypotenuse) that in music, as in geometry,
there are fractions, m/n, that are incommensurables
– such as the square root of 2, which cannot be expressed with
whole numbers or fractions, the body of rational numbers, but with
irrational numbers not yet developed. This discovery was held as
a secret among the Pythagoreans and led to the separation of algebra
and geometry for centuries, until Descartes in the 17th century
united them again.

For music it meant that
there was no center of an octave, no halving of the whole tone, no
perfect union of opposites, no ‘rationality’ to the
cosmos. The semitone could be the
‘door’ to other dimensions!

My task here is to give
some clues to the meta-physical functions of semitones, which seem to
involve the potential to shift to a different world or enter another
dimension. The key to attaining a different spiritual world exists in
the search for the exact right tone that resonates with that
particular ‘door’ to other dimensions and worlds. The
human being contains more dimensions than just three spatial
dimensions.

Philolaus

We have to bear in mind
that Pythagoras himself left no written record of his work; it was
and is against esoteric principles. Neither did those few students
who survived the pogrom of Pythagoras. It is one in the next
generation of Pythagoreans, Philolaus (ca.480- ? B.C.), who broke the
precept of writing down the master’s teaching. However,
Philolaus' records are lost, so it is Nicomachus’ fragments of
his writing, in his Manual of Harmonics, that is actually the only
source posterity has.

According to Nicomachus
/ Philolaus, the whole tone, 9/8, was divided differently from the
Pythagoreans method, by representing the whole tone with 27, the cube
of 3, a number highly esteemed by the Pythagoreans. Philolaus divided
the whole tone in two parts, calling the lesser part of 13 units a
‘diesis’, and the greater part of 14 units, ‘apotome’.
Philolaus had, in effect, anticipated Plato's calculations in the
Timaeus!

Timaeus
by Plato

Plato (427-347 B.C.)
gave in his work Timaeus a new meaning to the Pythagorean harmonic
universe by – in a purely mathematical method – enclosing
it within the mathematically fixed limits of four octaves and a major
sixth. It was determined by the numbers forming two geometrical
progressions, of which the last term is the twenty-seventh multiple
of the first term:27 = 1+2+3+4+8+9The two geometric
progressions in which the ratios between the terms is 2:1 and 3:1
are, respectively:1-2-4-8 and 1-3-9-27. Combining
these two progressions, Plato produced the seven-termed series:
1-2-3-4-8-9-27. The numbers in this series contain the octave, the
octave and a fifth, the double octave, the triple octave, the fifth,
the fourth and the whole tone. The entire compass from one to
twenty-seventh multiple comprises, therefore, four octaves and a
major sixth. In numerical terms it contains four octaves, 16:1 * 3:2
(a fifth) * 9:8 (a whole-tone) equals 27:1.Plato then proceeded
first to locate in each of the octaves the harmonic mean, the fourth,
then the arithmetic mean, the fifth. By inserting the harmonic and
the arithmetic means respectively between each of the terms in the
two geometric progressions, Plato formulated mathematically
everything Pythagoras had formulated by collecting acoustic data.

Plato did, however,
independently of the Pythagoreans, compute the semitone in the
fourth, which consists of two whole tones plus something, which is
less than the half of a whole tone, namely 256:243, the
leimma.According to Flora Levin in her commentary on Nicromachus'
”The Manual of Harmonics”, Plato went further than
Pythagoras by completing all the degrees in a diatonic scale:

1

9/8

81/64

4/3

3/2

27/16

243/128

2

E

F#

G#

A

B

C#

D#

E'

Plato's calculations led
to the inescapable fact of no center to the octave, no halving of the
whole tone with rational numbers, no rationality of the cosmos.
Nicomachus did his part in covering up the secret by
misrepresenting Plato and putting off some of the shattering
discoveries of irrational numbers to some future time.

The
semitones in the different modes

Pythagoras had practiced
music long before he transformed the heptachord into an octachord
that led him to discover the mathematical laws determining the basic
structure of an octave. He had fully understood the therapeutic value
of music in healing the body and soul. Most of all, he knew the set
of conditions for melody. He recognized strongly that every
tetrachord on which melody was based embodies the ‘natural’
or physical musical progression of whole tone-whole
tone-semitone. He maintained the fundamental structure of
both tetrachords in his scale, and for musical reasons he understood
that this distribution of intervals had to be maintained for all
melodic purposes with their configurations and inversions. This
was the foundation of the ancient Greek music, which further
developed into The
Greater Perfect System.

The
confusion of systems

The Greek music has an
inherent confusion of musical systems: a mix of the cyclic system of
perfect fifths (Pythagorean tuning), and the modal system
(tetrachords). We can only get a very faint idea of what ancient
Greek music really was about because European theorists through time
have made errors and misunderstandings.

In reality, the Arabs
and the Turks happened to receive directly the inheritance of Greece.
In many cases the works of Greek philosophers and mathematicians
reached Europe through the Arabs. Most serious studies on Greek music
were written by Arab scholars such as al-Färäbi in the
tenth century and Avicenna a little later, while Westerners -
Boethius in particular - had already made the most terrible
mistakes. It is the Arabs who maintained a musical practice
in conformity with the ancient theory, so to get an idea of ancient
Greek music, we should turn to Arab music.

The
Pythagorean Tuning

The musical
scale, said to be created by Pythagoras, was a diatonic musical scale
with the frequency rate as:

1

9:8

81:64

4:3

3:2

27:16

243:128

2.

This scale is
identical to the Chinese cyclic scale of fifths,
if we take F as the tonic. It has 5 major tones (9/8) and 2
semitones, leimma (256/243), in the mi-fa and si-do interval.The
third, 81/64, is a syntonic comma sharper than the harmonic third,
5/4.Here is the seven-notes Chinese
scale:

C

D

E+

(F)¤L+F#1/1

G

A+

B+

C'

1/1

9/8

81/64

4/3 (729/512)

3/2

27/16

243/128

2/1

¤
Let us note here that the most striking difference between the system
of fifths and that of harmonic relations to a tonic resides in the
perfect fourth, which is an essential interval in the scale of
proportions. The scale of fifths has an augmented fourth as its
sixth fifth, (3/2).

The Pythagorean scale
was based on the three prime intervals: the octave, the perfect 5th
and the perfect 4th. “Everything obeys a secret music of which
the ‘Tetractys’ is the numerical symbol”
(Lebaisquais).By generating 12 perfect fifths in the span of 7
octaves, 12 tones were produced. In order to place the tones within
one octave, the descending perfect 4th (the subdominant) was used,
and a 12-note chromatic scale was made.

He discovered what later
was called the Pythagorean comma, the discrepancybetween 12 fifths and 7
octaves gives (3:2)12> (2:1)7.Calculated through, it is:
129.74634 : 128 = 1.014. Or in cents: 23.5. Donot mistake
Pythagoras' Comma for the syntonic comma, equal to 22 cents, which is
derived from the difference between the major tone and the minor tone
in the Just Diatonic Scale, or discrepancy between the Pythagorean
third and the third in the harmonic series which is 5:4.

As far back as 2,500
years ago the Pythagorean figured out that it was impossible to
derive a scale in which the intervals could fit precisely into
an octave. The ancient Greeks explained this imperfection –
the comma – as an example of the condition of mortal humans in
an imperfect world. This fundamental problem with the 3 prime
ratios: 2:1, 3:2, 4:3 – which can be formulated in mathematical
terms as interrelated prime numbers having no common
divisor except unity – has been compromised in a number of
different temperaments of the diatonic scale up to our time.

In ancient Greek music
several other modes were used based on the tetrachords with a span of
the perfect fourth. Later, two tetrachords were put together with a
full tone in between so an octave was established. A number of
different modes were used in practical music performance. The
different placement of the two half tones made the different modes.
An
account of ancient Greek contributions to musical tuning would not be
complete without mentioning the later Greek scientist Ptolemy (2nd C.
A.D.). He proposed an alternative musical tuning system, which
included the interval of the major third based on that between the
4th and 5th harmonics, 5 / 4. This system of tuning was ignored
during the entire Medieval period and only re-surfaced with the
development of polyphonic harmony.

Gregorian
church music

From those ancient Greek
modes the Christian
Gregorian church derived its music,
though their names were a complete mix-up of the original Greek names
for their modes. What is important in this context is the placement
of the two semitones in the octave. They were placed differently in
order to create different modes that produced a special tonality or
mood. The interaction between tones and semitones made each
characteristic mode. The Gregorian church music from the late
Middle Ages developed an amazing beauty and spirituality. We owe the
monks – and Hildegard
von Bingen - a debt of gratitude for their part – singing
to worship the refinement of the soul and Divinity. A side effect
was the healing power in the strong proportion of higher harmonics,
which invigorating effect Alfred Tomatis has described in my page The
Power of Harmonics.

As long as musical
practice was mainly monophonic, the number of scales could be many.
When the wish for harmonious polyphonic singing was appearing, the
elimination of scales began because only the scales that were in
agreement with the harmonics could be used.

Polyphonic
music

The development in
musical practice from monophonic to polyphonic, and after the
Renaissance (the end of 15th century) to harmony, made it necessary
to have especially the third harmonized.
The Pythagorean third (81:64) is a syntonic comma larger than the
harmonic third (5:4). The need for harmonizing the third in the
part-songs became imperative as the polyphonic music became
predominant.

Just
Intonation - a scale of proportion

Since the major triad
became the foundation of harmony in
Western music, the Pythagorean scale has largely been discarded in
favor of the Just Diatonic Scale, or the scale of Zarlino
(1540-94). The frequencies of the notes in a root position
major triad are given by the fourth, fifth and sixth harmonics in the
harmonic series, i.e. the frequencies should be in the ratio 4: 5: 6.
(1-5:4-3:2).

The
Major Triad as a generator

If we look at this triad
as C, E, and G, the tonic triad, and associate it with its dominant
G, B, D and the tonic’s sub-dominant F, A, C, each of which has
one tone in common with the triad of the tonic, we obtain the
complete series of tones for the major scale of C:

1

9:8

5:4

4:3

3:2

5:3

15:8

2.

This scale consists of
three different intervals: major tone 9/8, minor tone 10/9, and major
half tone 16/15. Therefore, when the tonic is changed, we shall
obtain sharps and flats of different nature in order to keep the
frame of the scale, and the very notes of the original scale will in
some cases have to be raised or lowered by one comma (the difference
between the major and the minor tone).Those
who are familiar with Rodney Collin's The Theory of Celestial
Influence will notice that the Just Intonation is the scale he
applies to his great work on octaves by multiplying by 24.

The scale of Zarlino
(Just Intonation) is basically a mix of notes generated by fifths,
which allows right transpositions and notes which make correct
harmonic intervals; so in practice, two different systems are used
conjointly, which results in awkward transpositions.

The
Mean Tone

In musical practice,
especially when playing with key-instruments or the simple modulation
of keys, the Just Intonation causes many difficulties, mainly due to
the fact of the major and minor tones. The two different intervals of
a tone in this scale was for that reason modified during the 17th
century into a mean or average of the major and minor tone. Since
these two tones together equal a major third, the mean tone is equal
to half of the major third, or 193 cents.

This temperament is not
surprisingly called Mean Tone temperament, or 1:4 comma mean tone
(the fifths are all equal, but have been tempered by 5.5 cents, a
quarter of a syntonic comma) and was the most used temperament in
Baroque music.

There were some problems
with the enharmonic notes. The two diatonic semitones do not add up
to give a (full) tone. The Mean Tone semitones are 117.5 cents. So if
one wishes to play in more than six major and three minor keys, there
is trouble. This is because en-harmonically equivalent notes will not
have the same frequency. Additionally, this temperament has some real
false notes, called ‘wolf notes’, due to the 3.5 cents
short fifth, so the circle will fall short of closure by 12 x 3.5
cents = 42 cents.

Equal
Temperament

The ultimate compromise
appears in Equal Temperament, which is a circular
temperament. The Pythagorean comma (as approximately 24 cents) made
the circle too large. If the 12 perfect fifths –702 cents –
are equally distributed but contracted with 2 cents each, the circle
of fifths will be complete into a circle.In
the late 17th and 18th centuries a number of circular temperaments
were employed making use of this device. It is often said that J.S.
Bach's 48 Preludes and Fugues were written to demonstrate the
effectiveness of Equal Temperament. However, recent research (Barnes
1979) has shown that he probably wrote them for a circular
temperament similar to one devised by Werckmeister (known as
Werckmeister III), where the distribution of fifths was unequal; some
were 6 cents smaller, some were perfect.

The Equal Temperament as
we know it is completely equally
distributed, slightly diminished fifths (700 cents), that at one blow
eliminates the question about different frequencies of the enharmonic
notes and modulation limitations. The octave is equally divided
into 12 semitones of 100 cents. The frequency ratio for each of the
semitones is the twelfth root of an octave: (2/1) 1/12 =
1.059463094.../1.

This temperament has two
scales, a major and a minor. The difference lies in the third, sixth
and seventh, which are a half tone lower in the minor scale. Note
that the same intervals are present in the minor scale as in the
major scale, although the order is different.We
will not deal with the harmonic minor scale or the melodic minor
scale.

Let us make a comparison
in cents of the above mentioned scales with the Just Diatonic Scale
(Just Intonation) as base:

Scale

C

D

E

F

G

A

B

C'

Just
Diatonic

0

0

0

0

0

0

0

0

Pythagorean

0

0

+22

0

0

+22

+22

0

Mean
Tone

0

-11

0

+5.5

-5.5

+5.5

-5.5

0

Equal
Temperament

0

-4

+14

+2

-2

+16

+12

0

In
the western culture Equal Temperament is now so established and its
tonality so tuned in our ears, that it sounds just right, though the
third and the sixth ought to give problems because they are pretty
much sharper than the much purer and expressive Just Intonation.

The
artificial Equal Temperament

The great German
scientist from the 19th Century, Hermann von Helmholtz, who was also
a capable musician, made a strong stand for the Just Intonation
scale. He claimed in his On the Sensations of Tone that... “continual
bold modulational leaps threaten entirely to destroy the feeling for
tonality.” Further he states: “The music based on the
tempered scale must be considered as an imperfect music... If we
suppose it or even find it beautiful, it means that our ear has been
systematically spoiled since childhood.”

Professor Helmholtz
brought many examples of beautiful use of Just Intonation in singing
by use of the English system ‘Tonic Sol Fa-ists’, which
overcame the difficulties of modulation by using a different musical
notation system. Strings and wind instruments could also perform
this; so can modern keyboards.The
discussions about Equal Temperament versus Just Intonation have
continued up to present time. Daniel White has on his web page Tuning
& Music Scales Theory made an in-depth analysis of this
matter, concluding that ET sounds ‘sweeter’ than JT.

Compared with the other
scales we have gone over, the Equal Temperament has no definite
relations between the sounds since it has lost its relationship with
simple ratios. The more complicated the ratios are, the more
dissonant are the chords. We have been used to the muddy sounds, but
people in the East who are trained in modal memory and clear harmonic
relations cannot conceive the meaning of Western music. The
Equal Temperament has, in spite of its obvious weakness, made it
possible for great composers to create beautiful music with
extraordinary numbers of new chords and modulations. In the
twentieth Century the tendency to move away from simple ratios
between notes to sound ratios even far away from Equal Temperament
became manifest in atonal music.

Modern
Dodecaphonic music

In modern times a number
of atonal scales has been developed to serve the new dodecaphonic
music (Schönberg, Berg, Webern), where the classical notion of
harmony and rhythm is dissolved. Basically, the ancient diatonic
scale with its five whole tones and two semitones has been replaced
with a pure chromatic scale, which is a main factor in the change
from melodic tonal music to atonal dodecaphonic music. Though I
am very fond of non-figurative art, the modern atonal music is still
difficult for me to enjoy spontaneously. Educated people assure me of
the new beauty in contemporary music, which I can hear with my head,
but not with my heart. I have, however, observed a certain
indifference in the mainstream of classical music, and find myself
attracted to the early European music and folk (World) music.

My
main objection to the atonal dodecaphonic music lies in its
detachment from the physical world. The scale belongs to the
invisible realm because it is created by ratios far away from the
small numbers, which are related to the perceptible world and basic
emotions. In the ancient musical systems we have seen how
closely the musical scale had to be related to the perceptible world
represented mainly as small numbered ratios (low number of generating
fifths in the cyclic system or simple harmonic ratios in the modal
system).

Cyclic
and modal numbers

In this world of five
elements in which we live, no prime number higher than five can enter
into a system of sounds representing melodic or harmonic relations.
The Chinese system of cyclic fifths even refuses to get beyond this
number five; all its intervals are expressed in terms of powers of
two or three. The number for cyclic systems is three.Some modern
theorists are using the terms ‘3
limit scale’and ‘5 limit scale’.

The introduction of the
factor of five brings us to the harmonic modal scale, of which the
characteristic intervals are the harmonic major sixth, 5/3, the
harmonic major third, 5/4, the minor third, 6/5, the major half tone,
16/15 (24 /3*5), the minor half tone, 25/24, (52 /3*23 ) the syntonic
comma, 81/80, (34 / 24 * 5), and so forth.Compared with the Equal
Temperament, the tempered half tone is something like 1,059,463,094 /
1,000,000,000 against the major harmonic half tone 16/15.

The number five
‘humanizes’ the music. It makes the music an instrument
of expression of tangible reality. The introductions of higher prime
numbers, such as seven, would take us beyond this reality into
regions that are not within the scope of our normal perceptions and
understanding.Seven is considered the number of heavenly as well
as infernal regions. We have actually no means of knowing to which
side it may lead us!

In my opinion
you can only touch humans deeply if you play harmonious or tonal
music, because these tones belong to the real world and the man who
walks the Earth. The scale has to be more or less in accordance with
the lower harmonics in the series. The
way we hear and analyze sound is actually much the same as the
standing wave in a string. The basilar membrane in the inner ear
behaves like a ‘string’, and the ‘software’
in the brain is designed to look for the harmonic series. It is the
most agreeable - and most basic. What it all comes down to is that
the only measure for all phenomena is the human.

Reference
tone

Before a concert begins,
a reference tone, the concert pitch, is played so the instruments can
tune their middle a'. In modern times the pitch was set to 440 Hz. by
the second International Standard Pitch Conference in London 1938. It
is a high pitch compared to the older concert pitch of 435 Hz., which
was introduced by the French government in 1859 in cooperation with
musicians such as Hector Berlioz, Meyerbeer and Rossini.The
concert pitch has varied in earlier times, depending on country and
time. In the book On the Sensation of Tone by Helmholtz, a record of
concert pitch in Europe covers many pages. The characteristic for
Western music is that concert pitch is arbitrary.
It has no relation to forces above man. There is no reference to
earthly or celestial influx, but only to an artificial standard.

For the old Chinese the
tuning of their fundamental tone, Kung, was a matter of utmost
importance for their civilization; it had to be in alignment with the
Cosmic tone so the celestial influence could be channeled into
society by music.

We earlier mentioned
Cousto's calculations (in his book The Cosmic Octave). He relates the
Kung to the frequency of the Platonic Year. The note of the Platonic
year is found to be F in Western Equal Temperament pitch, which is in
the 48th octave with a frequency of 344.12 Hz.

The Indians’ method
had the character of meditation, since the musician not only has to
tune his instrument to the keynote in the prelude, he also attunes
himself to it, and gives the audience the opportunity to do so too.
This long introduction is essential since the musicians have to tune
in to the ‘sadja’, the everlasting, never-ceasing tone.
According to Indian tradition it stands for primordial vibration,
which is called ‘nada’ and expresses the universal OM.

The OM sound, according
to Cousto, corresponds approximately to the C sharp in the small
octave of the present day tuning system (136 Hz), and to the 32nd
octave tone of the Earth year. It means that in lowering 136 Hz tone
by 32 octaves, the resulting frequency will be as slow as the amount
of time it takes the Earth to circle the sun.

It is interesting to
note that the Indians arrived at this tone, which we can calculate
mathematically, ‘simply’ through intuition and
meditation.(The
calculation is: A day consists of 86,400 seconds. A tropical year has
365,242 days = 31,556,925,9747 seconds. The reciprocal value
multiplied by 232 = 136,10221 Hz.)Concert
pitch in western music, which is 440 Hz for the middle A, ought to be
435,92 Hz based on the note corresponding to the average solar day,
according to Cousto.

“It
don't mean a thing, if it ain't got that swing”Those
who are familiar with the jazz (swing) musician, Duke Ellington, will
‘hear’ Ella Fitzgerald sing this song. The reason I will
end on that note is to make clear that music is more than scales and
right tuning. Music contains of four major elements: Melody,
Rhythm, Harmony and Interpretation or Intention.

Having this in mind I
will continue with The
Sound of Silence, where I will extend the law of octaves into
realms other than scales and tuning by an elaboration on the
metaphysical properties of sound and music.